How to make a edge of an object curved (I don't think I'm saying that right but please help!)
You could need both subsurf and/or bevel, depending on what you need.

Here's an example, starting from a box extruded and shaped to a similar starting object:



the edge trick is shown on the right wireframe: adding edges near subsurfed rounded corners, you get somewhat sharp edges if you need them, but leaving subsurf for the rest of the mesh...

Hth